Functional analysis is a subfield of
mathematics which studies
problems in
analysis, including
differential equations,
integral equations,
Brownian motion and
probability, by
thinking of the
functions in question as
elements of
vector spaces, and studying the behavior of the
linear mappings or
operators which connect them. In this sense
functional analysis is an
infinite-dimensional generalization of
linear algebra.
In a more subtle sense, functional analysis can be thought of as a generalization of calculus in which multiplication is not always commutative, since number-valued functions are replaced by operator-valued ones. In this form it is needed to state the theory of quantum physics, since at the quantum level, the order in which measurements are performed can affect the results.
Objects and ideas connected with functional analysis include Banach spaces, Hilbert spaces, linear operators, spectra, and the Lebesgue integral.